example1_00 - Examples(Application of Ficks first law to...

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Examples (Application of Fick’s first law to steady state problems with constant diffusion coefficient (problem 1 and part of problem 3), and Fick’s second law to non-steady problems with thin-film and constant surface concentration boundary conditions, and constant diffusion coefficient (problem 2 and part of problem 3)) Problem 1 A cubic steel tank of volume 1 liter and wall thickness 0.1 cm is used to store hydrogen with an initial pressure of 8.7 atm. The tank is placed in outer space at 673 K. The hydrogen concentration on the steel surface is given by cp H = 3 (ppm) where c is in ppm and the hydrogen pressure, p H , is in atm. The density of steel is 7.73 g/cm 3 . The diffusion coefficient of hydrogen in steel is taken to be 10 -8 cm 2 /s. What is the rate of pressure drop (atm/s) as a result of diffusion of hydrogen through the wall? Solution: Assuming the process is diffusion-controlled, the outer surface concentration of the tank is fixed at zero at all times. The inner surface concentration is a function of hydrogen pressure inside the tank. The concentration in terms of g/cm 3 is then given by p p HH H × = × 32 3 2 (ppm) = 3 10 7.73 (g/cm 10 (g/cm -6 3 -5 3 ). ) Since the process is very slow and low, we can approximate this problem using the steady state Fick’s first law. JD dc dx p p H H =- × - - - 10 0 2 32 10 01 23 10 8 5 12 . . . The total flux of hydrogen diffusing out from the tank is given by JJ A p p p tot H H H ==× × (29 = × -- - 2 3 10 6 1000 1 38 10 6 9 10 12 1 3 2 91 0 .. . (g/s) mol/s From mass balance, the total flux out of the tank is the equal to the change in the amount of hydrogen per unit time within the tank, i.e., J dn dt tot = where n is the number of moles of hydrogen. Assuming hydrogen gas behave ideally, we have J dn dt V RT dp dt tot H == where V is the volume of the tank, R is gas constant and T is temperature. Therefore, dp dt RT V Jp p H tot H H × ×× 0 082 673 1 69 10 38 10 10 8 . . . (atm/s) It can be easily seen that the pressure dropping rate is proportional to the square root of the hydrogen pressure inside the tank. The initial pressure is 8.7 atm, so the initial rate of pressure decrease will be dp dt p H × = × 38 87 112 10 87 . . . . (atm/s) -------------------------------------------------------------------------------------------------------------------------
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Problem 2 There are two ways of introducing dopants into silicon by diffusion. The first is called predeposition during which the surface concentration of dopant atoms is maintained constant by a vapor source; and the second is called the redistribution or drive-in which is used to move predeposited dopants to the desired junction depth. a) Assume the substrate is n-type with a background doping level of 1.5x10 16 atoms/cm 3 and the diffusion coefficient of boron in silicon is given by D B ( T ) = 0.76exp - 3.46 eV k B T [ cm 2 / s ] where k B is the Boltzmann constant and T is temperature. The boron surface concentration is maintained at a constant value of 1.8x10 20 atoms/cm 3 during predeposition at 950˚C in a neutral ambient. If predeposition is performed for 30 min., calculate: (I) the diffusion profile; (II) the
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example1_00 - Examples(Application of Ficks first law to...

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